# American Institute of Mathematical Sciences

February  2012, 32(2): 353-379. doi: 10.3934/dcds.2012.32.353

## Lie's reduction method and differential Galois theory in the complex analytic context

 1 Universidad Sergio Arboleda, Calle 74 no. 14-14, Bogotá, D.C., Colombia 2 Departamento de Matemática e Informática Aplicadas a la Ingeniería Civil, ETSI Caminos, Canales y Puertos, Universidad Politécnica de Madrid, Profesor Aranguren s/n (Ciudad Universitaria) - 28040 Madrid, Spain

Received  December 2010 Revised  April 2011 Published  September 2011

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generalization of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot system is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. Lie. We propose an differential Galois theory for automorphic systems, and explore the relationship between integrability in terms of Galois theory and the Lie's reduction method. Finally we explore the algebra of Lie symmetries of a general automorphic system.
Citation: David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353
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##### References:
 [1] Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784 [2] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [3] Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701 [4] Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020 [5] Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065 [6] Fathalla A. Rihan, Yang Kuang, Gennady Bocharov. From the guest editors: "Delay Differential Equations: Theory, Applications and New Trends". Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : i-iv. doi: 10.3934/dcdss.2020404 [7] Yanzhao Cao, Anping Liu, Zhimin Zhang. Special section on differential equations: Theory, application, and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : i-ii. doi: 10.3934/dcdsb.2015.20.5i [8] Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304 [9] Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211 [10] Wenying Feng, Guang Zhang, Yikang Chai. Existence of positive solutions for second order differential equations arising from chemical reactor theory. Conference Publications, 2007, 2007 (Special) : 373-381. doi: 10.3934/proc.2007.2007.373 [11] Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020021 [12] Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75 [13] Santiago Capriotti. Dirac constraints in field theory and exterior differential systems. Journal of Geometric Mechanics, 2010, 2 (1) : 1-50. doi: 10.3934/jgm.2010.2.1 [14] Luigi Fontana, Steven G. Krantz and Marco M. Peloso. Hodge theory in the Sobolev topology for the de Rham complex on a smoothly bounded domain in Euclidean space. Electronic Research Announcements, 1995, 1: 103-107. [15] Miriam Manoel, Patrícia Tempesta. Binary differential equations with symmetries. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1957-1974. doi: 10.3934/dcds.2019082 [16] Guojie Zheng, Dihong Xu, Taige Wang. A unique continuation property for a class of parabolic differential inequalities in a bounded domain. Communications on Pure & Applied Analysis, 2021, 20 (2) : 547-558. doi: 10.3934/cpaa.2020280 [17] Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 [18] Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 [19] Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307 [20] Tomás Caraballo, Gábor Kiss. Attractivity for neutral functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1793-1804. doi: 10.3934/dcdsb.2013.18.1793

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